Stochastic calculus and SDEs
The Itô integral, Itô's lemma, local time and stochastic differential equations such as geometric Brownian motion.
3 lessons open · 3 planned
Steps 12, 13 and 14 are part of the option pricing path, a 22-step course from discounting to Black–Scholes.
Pages you can open
The Itô integral
How do you integrate against a Brownian path, which has no slope?
Itô's lemma
A Brownian path has no slope. What takes the place of the chain rule for \(f(W_t)\)?
Geometric Brownian motion
If a price moves by small random percentage changes, what does its distribution ten years from now look like?
Planned
In learning order. These are not pages yet.
- Itô isometry: the variance of a stochastic integralGraduate
- Simulating an SDE with the Euler-Maruyama methodAdvanced undergrad
- Local time: how long Brownian motion lingers at a levelGraduate
How it connects
Builds on Stochastic processes Calculus Numerical methods Probability
Leads to PDE methods Monte Carlo and numerical pricing Volatility and jump models Derivatives Arbitrage and risk-neutral pricing Hedging and calibration Differential equations Optimization Stochastic processes Equities and portfolios Interest-rate models
Subjects marked “connection” are linked by a related idea, not by a prerequisite.